1 Introduction
The foundations of electromagnetic theory were established in the nineteenth century to explain experimental evidence associated with a wide variety of phenomena concerning electric charge, its flow, and links with magnetic effects. In a two-volume treatise Maxwell in 1873 presented a meticulous and comprehensive survey [
1] of phenomena and experimentation, and developed a theoretical framework within which results could be interpreted. In a radical departure from the Newtonian concept of action at a distance, Maxwell’s account introduced relations between fields (that is, continuous functions of position and time) holding in the aether, on the basis of which he conjectured the electromagnetic nature of light. In particular Maxwell’s equations included (using modern notation: see Hendry [
2], Chap. 6)
$$\begin{aligned} \mathrm{div}\, \mathbf{D} =& \rho, \end{aligned}$$
(1.1)
$$\begin{aligned} \partial\mathbf{D}/\partial t + \mathbf{j} =& \mathrm{curl}\, \mathbf{H}, \end{aligned}$$
(1.2)
$$\begin{aligned} \mathbf{B} =& \mathrm{curl}\, \mathbf{A} \end{aligned}$$
(1.3)
and
$$ \mathbf{f} = \mathbf{v} \times\mathbf{B} - \partial\mathbf{A}/ \partial t - \nabla\psi. $$
(1.4)
Here
\(\mathbf{D}\) denotes electric displacement,
\(\rho\) charge density,
\(\mathbf{j}\) electric current density,
\(\mathbf{H}\) magnetic displacement,
B magnetic field,
A magnetic potential,
f electromotive force,
v velocity (relative to the aether), and
\(\psi\) electrostatic potential.
Immediately (
1.3) yields
$$ \mathrm{div}\, \mathbf{B} = 0. $$
(1.5)
Further, on taking the divergence of relation (
1.2) and the time derivative of (
1.1) it follows that
$$ \partial\rho/\partial t + \mathrm{div}\, \mathbf{j} = 0. $$
(1.6)
If the electric field
E is defined via
$$ \mathbf{E} := -\partial\mathbf{A}/\partial t - \nabla\psi, $$
(1.7)
then the time derivative of (
1.3) yields
$$ \partial\mathbf{B}/\partial t = - \mathrm{curl}\, \mathbf{E}, $$
(1.8)
and relation (
1.4) may be written as
$$ \mathbf{f} = \mathbf{v} \times\mathbf{B} + \mathbf{E}. $$
(1.9)
The relations which are now considered to be the most fundamental, and which bear Maxwell’s name, are (
1.1), (
1.2), (
1.5) and (
1.8).
Attempts to deduce Maxwell’s equations from dynamical principles were restricted by contemporaneous understanding of the nature of matter. In 1902 Lorentz [
3,
4] proposed a system of Maxwell-like equations which related to individual small isolated regions of charged matter he termed electrons. Each electron was characterised by a charge density field
\(\rho\), which vanished outside the region it occupied, together with fields of velocity
v, electric displacement
d, and magnetic force
h. In stationary aether these were to satisfy (cf. [
4], equations I, II, IV, V, VI)
$$\begin{aligned} \mathrm{div}\, \mathbf{d} = \rho, & \quad \quad \partial\rho/\partial t + \mathrm{div}\, \rho\mathbf{v} = 0, \end{aligned}$$
(1.10)
$$\begin{aligned} \mathrm{div}\, \mathbf{h} = 0, & \quad \quad \mathrm{curl}\, \mathbf{h} = c^{-1} ( \partial\mathbf{d}/\partial t + \rho\mathbf{v}), \end{aligned}$$
(1.11)
and
$$ \partial\mathbf{h}/\partial t = - c \,\mathrm{curl}\, \mathbf{d}, $$
(1.12)
where
\(c\) denotes the speed of light in the aether. Further, ‘the force, reckoned per unit charge, which the aether exerts on a charged element of volume’, was postulated (cf. [
4], equation VII) to be
$$ \mathbf{f} = \mathbf{d} + c^{-1} \mathbf{v} \times\mathbf{h}. $$
(1.13)
Here
\(\mathbf{d}\) was identified as ‘the electric force that would act on an immovable charge’. The foregoing fields were considered to change rapidly and irregularly, and observable/measurable quantities were to be identified with local spatial averages computed over many charges (cf. [
3]). Specifically, the average of any field
\(F\), evaluated at location
x, is
$$ \langle F\rangle(\mathbf{x}) := V^{-1} \int_{C(\mathbf{x})} F\, dv. $$
(1.14)
Here the volume integral is taken over the averaging region
\(C(\mathbf{x})\) of volume
\(V\). Relations (
1.10)–(
1.13) were averaged using properties
1
$$ \partial/\partial t \bigl\{ \langle F\rangle \bigr\} = \langle\partial F/ \partial t\rangle, \qquad\nabla \bigl\{ \langle F\rangle \bigr\} = \langle \nabla F \rangle. $$
(1.15)
Distinguishing between electrons associated with conduction, polarisation, and magnetisation, Lorentz obtained the macroscopic Maxwell Eqs. (
1.1), (
1.5), (
1.8), and a version of (
1.2) of the form
$$ \mathrm{curl}\, \mathbf{H} = \mathbf{C} \quad\mbox{with }\mathrm{div}\, \mathbf{C} = 0. $$
(1.16)
Identification of composite current
C with
\(\partial \mathbf{D}/\partial t + \mathbf{j}\) yields (
1.2) and, via (
1.1) and (
1.16)
1, also (
1.6). Further, Lorentz effected decompositions
$$ \mathbf{D} = \mathbf{E} + \mathbf{P} \quad\mbox{and}\quad\mathbf{H} = \mathbf{B} - \mathbf{M}, $$
(1.17)
where
P and
M represented electric and magnetic polarisation densities.
Many studies have subsequently refined Lorentz’ pioneering work. In particular, once electrons and atomic nuclei were established to be the fundamental and discrete carriers of charge, it became natural to model these entities as point charges, and to attempt to introduce such knowledge into the microscopic relations. Further, averaging procedures, together with modelling assumptions concerning subatomic behaviour, have been clarified. Specifically, averaging may be purely spatial (Van Vleck [
6]), jointly in space and time (Rosenfeld [
7]), or statistical (via designation of an appropriate ensemble: Mazur and Nijboer [
8]). These approaches were reviewed by de Groot [
9] who also discussed covariant derivations. A more rigorous approach to spatial averaging via the introduction of a weighting function was employed by Russakoff [
10].
The aforementioned works are based upon the microscopic relations (
1.10)–(
1.13), even if atomicity is introduced via expressions for
\(\rho\) and
\(\mathbf{j}\, (:= \rho\mathbf{v})\) in terms of sums involving
\(\delta\)-functions and instantaneous point charge locations and velocities. Such atomicity is somewhat at variance with the interpretations of
d and
h, no matter how locally these fields are defined. It is helpful to bear in mind relevant spatial scales. The respective sizes of nuclei and atoms are of orders
\(10^{-15}\mbox{ m}\) and
\(10^{-10}\mbox{ m}\), and so the Lorentz equations, postulated to model electromagnetic behaviour within atoms, relate to behaviour at scale
\(10^{-10}\mbox{ m}\) or less. On the other hand Maxwell’s equations pertain to reproducible macroscopic behaviour at scales often described as ‘physically infinitesimal’ but ‘microscopically large’. Practically speaking, field values must be related to measurements, and hence to spatial and temporal sensitivity of monitoring devices. In any specific physical context it is thus for experimentation to determine the scales of length and time at which Maxwell’s relations provide a valid description.
Here the approach is from the outset entirely atomistic and no appeal is made to postulated microscopic relations. Electrons and atomic nuclei are modelled as point charges. Spatial averaging of the kinetic behaviour of any set of charges is effected in terms of a scale-dependent weighting function
\(w\). The starting point in Sect.
2 is the definition of a charge density field
\(\rho_{w}\) whose time derivative immediately introduces a current density
\(\mathbf{j}_{w}\) which satisfies (
1.6). Any solution
a to
\(\mathrm {div}\, \mathbf{a} = w\) leads directly to relations of forms (
1.1) and (
1.2), with specific definitions of the electrokinetic fields
\(\mathbf{D} _{w}\) and
\(\mathbf{H}_{w}\). A natural choice of
\(w\) is introduced in Sect.
3 which accords equal weighting to charges within a prescribed distance
\(\epsilon\) from any point
x at which an average is to be computed, and zero weighting to charges further than
\(\epsilon + \delta\) from
x, with
\(\delta\ll\epsilon\). For distances in the range
\([\epsilon, \epsilon+ \delta]\) weighting corresponds to a choice of mollifier which ensures that
\(w\) is everywhere smooth. The assumption that the corresponding function
a be isotropic leads to natural decompositions
\(\mathbf{D}_{w} = \mathbf{P}_{w} + \boldsymbol {\mathcal {E}} _{w}\) and
\(\mathbf{H}_{w} = - \mathbf{M}_{w} + \boldsymbol {\mathcal {B}}_{w}\), together with potentials
\(\psi_{w}\) and
\(\boldsymbol {\mathcal {A}}_{w}\) for which
\(\boldsymbol {\mathcal {E}}_{w} = - \nabla\psi_{w}\),
\(\boldsymbol {\mathcal {B}}_{w} = \mathrm{curl}\, \boldsymbol {\mathcal {A}}_{w}\) and
\(\partial\psi_{w}/\partial t + \mathrm{div}\, \boldsymbol {\mathcal {A}}_{w} = 0\). The contributions to values
\(\boldsymbol {\mathcal {E}}_{w} (\mathbf{x})\) and
\(\boldsymbol {\mathcal {B}}_{w} (\mathbf{x})\) from any charge
\(q_{i}\) distant further than
\(\epsilon+ \delta\) from
x are shown to be
\(-q_{i} \mathbf{u}_{i}/4\pi u^{3}_{i}\) and
\(q_{i} \mathbf{u}_{i} \times\mathbf{v}_{i}/4\pi u^{3}_{i}\), respectively. (Here
\(\mathbf{u}_{i}\) denotes the displacement of
\(q_{i}\) from
x,
\(u_{i}\) its magnitude, and
\(\mathbf{v}_{i}\) its velocity.) Additional time averaging is introduced in Sect.
4 to elucidate the physical interpretations of fields by considering bound and free/diffusive electrons in simple systems. In particular,
\(\mathbf{P}_{w}\) is seen to be a density of a measure of time-averaged electron charge distribution about parent nuclei: individual contributions are time-averaged dipole moments which measure orbital asymmetry. In Sect.
5 the electrokinetic fields
\(\boldsymbol {\mathcal {E}}_{w}\) and
\(\boldsymbol {\mathcal {B}}_{w}\) are linked with the force-related electrostatic and magnetostatic fields
\(\mathbf{E}^{s}_{w}\) and
\(\mathbf{B}^{s}_{w}\) via the experimental results of Coulomb and Biot-Savart, respectively. Specifically,
\(\mathbf{E}^{s}_{w} = \epsilon_{0}^{-1} \boldsymbol {\mathcal {E}} _{w}\) and
\(\mathbf{B}^{s}_{w} = \mu_{0} \boldsymbol {\mathcal {B}}_{w}\), where
\(\epsilon_{0}\) and
\(\mu_{0}\) are determined by experiment, satisfy
\(\epsilon_{0} \mu_{0} = c^{-2}\), and serve to ensure dimensional consistency. A formal generalisation to dynamical contexts motivated by (
1.7) leads to a complete set of Maxwell relations, consequent upon knowledge of the instantaneous location and velocity of every charge. However, such global instantaneous information can never be known, but requires time to be communicated. This issue is addressed in Sect.
6. It is assumed that information is transferred at the local speed of light and is visualised via the artifice of hypothetical radar signal reflection. The consequent fully-dynamical Maxwell relations correspond to knowledge of the apparent locations and velocities as monitored at any given location and time. Such information has required time to be transmitted and corresponds to an earlier (retarded) time. In Sect.
7 the consequences of force relation (
1.9) holding in a general dynamical context are explored. The individual contribution
\(\mathbf{B}^{d}_{i}\) of a charge
\(P_{i}\) to the dynamic magnetic field at location
x is shown to be orthogonal both to the corresponding dynamic electric field contribution
\(\mathbf{E}^{d}_{i}\) and to the apparent displacement of
\(P_{i}\) from
x. The force
\(\mathbf{F}_{i}\) on a charge at
x due to
\(P_{i}\) is expressed as a linear transformation acting on
\(\mathbf{E}^{d}_{i}\). A brief summary and concluding remarks are appended in Sect.
8.
The notation employed is direct (that is, free of co-ordinate considerations: cf. [
5]) and standard identities involving scalar and vector fields are employed without reference.
2 Spatial Averaging and Weighting Functions
Electromagnetic phenomena derive from the behaviour of electrons and atomic nuclei which are here modelled as point charges. Relations which describe spatially-averaged kinetic behaviour of assemblies of such charges are derived: these are formally identical to Maxwell’s Eqs. (
1.6), (
1.1) and (
1.2).
Here
\(m_{i}\),
\(q_{i}\) and
\(\mathbf{x}_{i} (t)\) denote the mass, charge, and location at time
\(t\), of a typical charge
\(P_{i}\). For any assembly of charges, the net charge within any region ℛ at time
\(t\), divided by the volume
\(V\) of
\(R\), yields a volumetric average (that is, a
density)
\(\rho(\mathcal{R}, t)\) at time
\(t\). Symbolically,
$$ \rho(\mathcal{R}, t) := \sum_{i}^{\prime} q_{i}/V, $$
(2.1)
where the primed sum is taken only over those charges in ℛ at time
\(t\). Equivalently,
$$ \rho(\mathcal{R}, t) = \sum_{i} q_{i} w_{i} (t), $$
(2.2)
where the sum is over
all assembly charges, and
\(w_{i} (t) = 1\) or 0 according to whether or not
\(P_{i}\) lies in ℛ at time
\(t\).
The weighted sum in (
2.2) can be generalised to yield a candidate electric charge density
field via
$$ \rho_{w} (\mathbf{x}, t) := \sum_{i} q_{i}\, w \bigl(\mathbf{u}_{i} ( \mathbf{x}, t) \bigr), $$
(2.3)
where the
displacement of
\(P_{i}\) from location
x at time
\(t\)
$$ \mathbf{u}_{i} (\mathbf{x}, t) := \mathbf{x}_{i} (t) - \mathbf{x}. $$
(2.4)
Suppressing arguments, (
2.3) may be written as
$$ \rho_{w} = \sum_{i}\, q_{i} w(\mathbf{u}_{i}). $$
(2.5)
Here
weighting function
\(w\) is defined on the space
\(\mathcal{V}\) of all displacements in three-dimensional Euclidean space, takes real values with physical dimension
\(L^{-3}\), and assigns greater values to charges near
x than those far therefrom. In order that the integral of
\(\rho_{w}\) over all space should yield the total net assembly charge it is sufficient that
$$ \int_{\mathcal{V}} w(\mathbf{u}) d\mathbf{u} = 1. $$
(2.6)
Consideration of an assembly consisting of a single charge indicates that normalisation condition (
2.6) is also necessary: cf. [
5], p. 45.
Remark
2.3 enables (
2.25) to be written as
$$ \partial\mathbf{D}_{w}/\partial t + \mathbf{j}_{w} = \, \mathrm{curl} \, \mathbf{H}_{w}, $$
(2.33)
where (cf. (
1.2)) candidate
magnetic displacement field
$$ \mathbf{H}_{w} := \sum_{i} q_{i}\, \mathbf{v}_{i} \times\mathbf{a} _{i} = \sum_{i} q_{i} \mathbf{v}_{i} \times\mathbf{a} (\mathbf{u} _{i}). $$
(2.34)
While relations (
2.25) and (
2.33) are equivalent, the former requires no appeal to orientation (that is, to ‘right-’ and ‘left-handedness’) which is necessary both for the definition of a vector product and of the curl operator.
5 Consequences of the Coulomb and Biot-Savart Laws
If
\(\boldsymbol {\mathcal {E}}_{i}\) denotes the contribution to
\(\boldsymbol {\mathcal {E}}_{w}\) of a charge
\(q_{i}\) for which
\(u_{i} > \epsilon+ \delta\), then (cf. (
3.37)
1 and (
3.34)
2) for any charge
\(q\)
$$ q\boldsymbol {\mathcal {E}}_{i} = -qq_{i}\, \mathbf{u}_{i}/4 \pi u^{3}_{i}. $$
(5.1)
This may be compared with Coulomb’s law for the force
\(\mathbf{f}^{es} _{qq_{i}}\) exerted
in vacuo on a stationary charge
\(q\) at location
x by a stationary charge
\(q_{i}\) at
\(\mathbf{x}_{i}\), namely (in SI units: cf., e.g., Griffiths [
11], (2.1))
$$ \mathbf{f}^{es}_{qq_{i}} = - qq_{i} \mathbf{u}_{i}/4\pi\epsilon_{0} u ^{3}_{i}. $$
(5.2)
Thus, for
in vacuo separations in excess of
\(\epsilon+ \delta\),
$$ \mathbf{f}^{es}_{qq_{i}} = q \boldsymbol {\mathcal {E}}_{i}/ \epsilon_{0}. $$
(5.3)
Accordingly, for any set
\(\{q_{i}\}\) of stationary charges all of which are distant at least
\(\epsilon+ \delta\) from
x, the assumption of linear superposition (cf. [
11], 2.4; Zangwill [
12], (2.18); Jackson [
13], 24–26; Elliott [
14], Sect. 3.2) yields a
static electric field
\(\mathbf{E}^{s}_{w}\) at
x for which
$$ q\mathbf{E}^{s}_{w} := \sum_{i} \mathbf{f}^{es}_{qq_{i}} = q \epsilon ^{-1}_{0} \sum_{i} \boldsymbol {\mathcal {E}}_{i}. $$
(5.4)
Hence, for such a collection of charges
$$ \mathbf{E}^{s}_{w} = \epsilon^{-1}_{0} \boldsymbol {\mathcal {E}}_{w}. $$
(5.5)
Similarly, if
\(u_{i} > \epsilon+ \delta\), then the contribution of charge
\(q_{i}\) to
\(\boldsymbol {\mathcal {B}}_{w}\) is (cf. (
3.39)
1 and (
3.34)
2)
$$ \boldsymbol {\mathcal {B}}_{i} := q_{i} \mathbf{u}_{i} \times \mathbf{v}_{i}/4 \pi u^{3}_{i}. $$
(5.6)
Consider a localised set of charges
\(P_{i}\) which lie within a sphere of radius
\(\epsilon\) centred at point
X. If
\(\mathbf{R} = \mathbf{X} - \mathbf{x}\) and
\(R := \Vert\mathbf{R}\Vert\gg\epsilon \), then
\(\mathbf{u}_{i} = (\mathbf{x}_{i} - \mathbf{X}) + (\mathbf{X} - \mathbf{x}) \sim\mathbf{R}\) and
$$\begin{aligned} \sum_{i} \boldsymbol {\mathcal {B}}_{i} \sim& \sum_{i} q_{i} \mathbf{R} \times \mathbf{v}_{i}/4\pi R^{3} \\ =& \bigl(\mathbf{R}/4\pi R^{3} \bigr) \times \biggl(\sum _{i} q_{i} \mathbf{v}_{i}/V _{\epsilon} \biggr) V_{\epsilon}\sim \bigl(\mathbf{R}/4\pi R^{3} \bigr) \times \mathbf{j}_{w} (\mathbf{X}) V_{\epsilon}. \end{aligned}$$
(5.7)
This may be compared with the Biot-Savart law in which the contribution
\(\Delta\mathbf{B}^{s}\) to the net magnetic field
\(\mathbf{B}^{s}\) at
x arising from such a collection of charges is (cf., e.g., [
12], (10.15)) essentially
$$ \Delta\mathbf{B}^{s} = \bigl(\mu_{0}\, \mathbf{R}/4\pi R^{3} \times \mathbf{j}^{s} \bigr) V_{\epsilon}. $$
(5.8)
Here
\(\mathbf{j}^{s}\) is the current density associated with the charges and is
steady (emphasised by superscript ‘
\(s\)’). Comparison of (
5.7) with (
5.8) suggests the natural identification (for the charges considered)
$$ \boldsymbol {\mathcal {B}}_{w} = \sum_{i} \boldsymbol { \mathcal {B}}_{i} =: \mu_{0}^{-1} \mathbf{B}^{s}_{w}. $$
(5.9)
As they stand, relations (
5.12) and (
5.13) do not of themselves indicate any restriction to macroscopically-static situations, and admit
formal generalisation to dynamical contexts without change. In particular, (
2.20) may be written, via (
5.11)
1, as
$$ \epsilon_{0}\, \mathrm{div}\, \mathbf{E}^{s}_{w} = \rho', $$
(5.27)
where
$$ \rho' := \rho_{w} - \mathrm{div}\, \mathbf{P}_{w}. $$
(5.28)
Further, (
2.33) may be expressed, via (
5.11)
1 and (
5.11)
2, as
$$ \epsilon_{0} \mu_{0}\, \partial\mathbf{E}^{s}_{w}/ \partial t + \mu _{0}\, \mathbf{J}' = \mathrm{curl}\, \mathbf{B}^{s}_{w}, $$
(5.29)
where
$$ \mathbf{J}' := \mathbf{j}_{w} + \partial \mathbf{P}_{w}/\partial t + \mathrm{curl}\, \mathbf{M}_{w}. $$
(5.30)
Thus
$$\begin{aligned} \partial\rho'/\partial t =& \partial\rho_{w}/\partial t - \partial /\partial t \{\mathrm{div}\, \mathbf{P}_{w}\} = \partial \rho_{w}/ \partial t - \mathrm{div}\, \{\partial\mathbf{P}_{w}/ \partial t\}, \end{aligned}$$
(5.31)
$$\begin{aligned} \mathrm{div}\, \mathbf{J}' =& \mathrm{div}\, \mathbf{j}_{w} + \mathrm{div} \{\partial\mathbf{P}_{w}/\partial t\}, \end{aligned}$$
(5.32)
and hence, via (
2.8),
$$ \partial\rho'/\partial t + \mathrm{div}\, \mathbf{J}' = 0. $$
(5.33)
While (
5.27) and (
5.29) are two often-cited versions of Maxwell relations (cf., e.g., Griffiths [
11], §10.1.1) the foregoing approach does not yield a corresponding version of (
1.8). Guided by (
1.7), suppose
$$ \tilde{\mathbf{E}}_{w} := \mathbf{E}^{s}_{w} - \partial\mathbf{A} ^{s}_{w}/\partial t = - \nabla \psi^{s}_{w} - \partial\mathbf{A}^{s} _{w}/\partial t. $$
(5.34)
Then clearly
$$ \mathrm{curl}\, \tilde{\mathbf{E}}_{w} = - \mathrm{curl} \bigl\{ \partial \mathbf{A}^{s}_{w}/\partial t \bigr\} = -\partial/ \partial t \bigl\{ \mathrm{curl}\, \mathbf{A}^{s}_{w} \bigr\} , $$
(5.35)
whence, from (
5.12)
2,
$$ \mathrm{curl}\, \tilde{\mathbf{E}}_{w} = - \partial \mathbf{B}^{s} _{w}/\partial t. $$
(5.36)
While this relation has been an immediate consequence of (
5.34), relation (
5.29), with
\(\mathbf{E}^{s}_{w}\) equated with
\(\tilde{\mathbf{E}}_{w} + \partial\mathbf{A}^{s}_{w}/\partial t\), introduces an extra term
\(\epsilon_{0} \mu_{0}\ \partial^{2} \mathbf{A}^{s}_{w}/\partial t^{2}\). Rather than adopting this approach, consider the direct consequence of definition (
5.34) which is summarised in the following result.
6 Signal Transmission Times, Retardation and Classical Maxwellian Electrodynamics
Consider the behaviour of a moving point charge \(P_{i}\) as monitored by an observer \(O\) located at a point x in an inertial frame ℱ. Information available to \(O\) at time \(t\) can involve only data that has reached \(O\) at, or before, this time. Since transmission of information is not instantaneous, it is necessary to examine the consequences of transmission time delay. Examined here are the relationship between the apparent location of \(P_{i}\) at time \(t\), the delay such datum of information takes to reach \(O\), and the actual trajectory of \(P_{i}\).
It is instructive to consider how, at least in principle, information about the motion of
\(P_{i}\) could be obtained. Suppose that
\(O\) has a radar device capable of detecting a reflected segment (from
\(P_{i}\)) of any signal that it has transmitted. Suppose further that any such segment, both during its outward and inward paths, travels in a straight line at constant speed
\(c\), and is instantaneously reflected by
\(P_{i}\). Such a signal, emitted from
x at time
\(t'\) and received back at time
\(t\), will have travelled a total distance
\(c(t-t')\) and been reflected at time
\(t' + (t-t')/2 = (t + t')/2\) when at a distance
\(c (t-t')/2\) from
x. If
\(\mathbf{x}_{i} ( \tau)\) denotes the location of
\(P_{i}\) at time
\(\tau\), then the distance travelled by the signal between its reflection and reception is
$$ \bigl\Vert \mathbf{x}_{i} \bigl( \bigl(t + t' \bigr)/2 \bigr) - \mathbf{x}\bigr\Vert = c \bigl(t - t' \bigr)/2. $$
(6.1)
Changing notation, if
\(\tau_{i} (\mathbf{x}, t)\) denotes the time at which the signal reaching
x at time
\(t\) was reflected from
\(P_{i}\), then
$$ \tau_{i} (\mathbf{x}, t) = \bigl(t + t' \bigr)/2, $$
(6.2)
a detectable quantity. Accordingly, (
6.1) may be written as
$$ \bigl\Vert \mathbf{x}_{i} \bigl(\tau_{i} (\mathbf{x}, t) \bigr) - \mathbf{x}\bigr\Vert = c \bigl(t- \tau_{i} (\mathbf{x}, t) \bigr). $$
(6.3)
Assuming that the device can detect the direction of the incoming signal, such information, together with (
6.3), determines location
\(\mathbf{x}_{i} (\tau_{i}(\mathbf{x}, t))\). This is the
apparent location of
\(P_{i}\) at time
\(t\) as monitored at location
\(\mathbf{x}, \mathbf{x}^{a}_{i} (\mathbf{x}, t)\) say. That is,
$$ \mathbf{x}^{a}_{i} (\mathbf{x}, t) := \mathbf{x}_{i} \bigl(\tau_{i} ( \mathbf{x}, t) \bigr) = ( \mathbf{x}_{i} \circ\tau_{i}) (\mathbf{x}, t). $$
(6.4)
Remark 6.1
In any motion of
\(P_{i}\), apparent location
\(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\) is unique. Indeed, suppose that a signal emitted from
x gives rise to segments which are reflected by
\(P_{i}\) at times
\(\tau\) and
\(\tau'\), and are both received at
x at time
\(t\). Thus
\(\tau\) and
\(\tau'\) are two values of
\(\tau_{i} (\mathbf{x}, t)\), and
\(\mathbf{x}_{i} (\tau)\) and
\(\mathbf{x}_{i} (\tau')\) are both candidates for
\(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\). From (
6.3) it follows that
5
$$\begin{aligned} \bigl\Vert \mathbf{x}_{i} (\tau) - \mathbf{x}_{i} \bigl( \tau' \bigr)\bigr\Vert =& \bigl\Vert \bigl(\mathbf{x}_{i} (\tau) - \mathbf{x} \bigr) - \bigl(\mathbf{x}_{i} \bigl(\tau' \bigr) - \mathbf{x} \bigr)\bigr\Vert \\ \ge& \bigl|\bigl\Vert \mathbf{x}_{i} (\tau) - \mathbf{x}\bigr\Vert - \bigl\Vert \mathbf{x}_{i} \bigl(\tau' \bigr) - \mathbf{x}\bigr\Vert \bigr| \\ =& \bigl|c(t-\tau) - c \bigl(t-\tau' \bigr)\bigr| = c\bigl|\tau-\tau'\bigr|. \end{aligned}$$
(6.5)
Hence the average speed of
\(P_{i}\) over a time interval of duration
\(|\tau- \tau'|\) is at least
\(c\), a physical impossibility. Thus the hypothesis of two times
\(\tau\) and
\(\tau'\) is incorrect,
\(\tau_{i} ( \mathbf{x}, t)\) is unique, and so
\(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\) is unique.
The
apparent displacement of
\(P_{i}\) from
x at time
\(t\) is
$$ \mathbf{u}^{a}_{i} (\mathbf{x}, t) := \mathbf{x}^{a}_{i} (\mathbf{x}, t) - \mathbf{x} = \mathbf{x}_{i} \bigl(\tau_{i} (\mathbf{x}, t) \bigr) - \mathbf{x}. $$
(6.6)
Writing
$$ u^{a}_{i} := \Vert\mathbf{u}^{a}_{i} \Vert, $$
(6.7)
relation (
6.3) becomes
$$ u^{a}_{i} (\mathbf{x}, t) = c \bigl(t - \tau_{i} (\mathbf{x}, t) \bigr). $$
(6.8)
Time
\(\tau_{i}\), apparent location
\(\mathbf{x}^{a}_{i}\), and apparent displacement
\(\mathbf{u}^{a}_{i}\) are functions of
x and
\(t\), and are thus
fields;
\(\tau_{i}\) is termed the
retarded time field. In particular,
\(\tau_{i} (\mathbf{x}, t)\) and
\(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\) constitute the basic information about any motion of
\(P_{i}\) which is available at
x at time
\(t\). Such information is necessary in computation of velocities. Here a distinction must be made between the
actual velocity of
\(P_{i}\) at any time
\(\tau\), namely
$$ \mathbf{v}_{i} (\tau) := d/d\tau \bigl\{ \mathbf{x}_{i} ( \tau) \bigr\} = \dot{\mathbf{x}}_{i} (\tau), $$
(6.9)
and the
apparent velocity
$$ \mathbf{v}^{a}_{i} (\mathbf{x}, t) := \partial/\partial t \bigl\{ \mathbf{x}^{a}_{i} (\mathbf{x}, t) \bigr\} = \partial/\partial t \bigl\{ \mathbf{u}^{a}_{i} (\mathbf{x}, t) \bigr\} $$
(6.10)
corresponding to
\(P_{i}\) as monitored at
x and time
\(t\). In particular,
\(\mathbf{v}_{i}\) is a vector-valued function of time, whereas
\(\mathbf{v}^{a}_{i}\) is a vector-valued field. From (
6.4) and the chain rule,
$$\begin{aligned} \mathbf{v}^{a}_{i} (\mathbf{x}, t) = \partial/\partial t \bigl\{ \mathbf{x}^{a}_{i} (\mathbf{x}, t) \bigr\} =& \partial/\partial t \bigl\{ \mathbf{x}_{i} \bigl(\tau_{i} (\mathbf{x}, t)\bigr) \bigr\} \\ =& \dot{\mathbf{x}}_{i} \bigl(\tau_{i} (\mathbf{x}, t) \bigr) \partial\tau_{i}/ \partial t. \end{aligned}$$
(6.11)
That is,
$$ \mathbf{v}^{a}_{i} (\mathbf{x}, t) = \mathbf{v}_{i} \bigl(\tau_{i} ( \mathbf{x}, t) \bigr) \partial\tau_{i}/ \partial t. $$
(6.12)
To determine
\(\partial\tau_{i}/\partial t\), note that from (
6.6) and (
6.8)
$$ \mathbf{u}^{a}_{i}\,.\, \mathbf{u}^{a}_{i} = \bigl(u^{a}_{i} \bigr)^{2} = c^{2} (t- \tau_{i})^{2}. $$
(6.13)
Differentiation with respect to
\(t\) yields, via (
6.10)
2 and (
6.8),
$$ 2\mathbf{u}^{a}_{i}\,.\, \mathbf{v}^{a}_{i} = 2c^{2} (t-\tau_{i}) (1- \partial\tau_{i}/ \partial t) = 2cu^{a}_{i} (1-\partial\tau_{i}/ \partial t). $$
(6.14)
Accordingly,
$$ \partial\tau_{i}/\partial t = 1 - c^{-1} \hat{ \mathbf{u}}^{a}_{i}\,. \, \mathbf{v}^{a}_{i} =: \beta_{i}, $$
(6.15)
where
$$ \hat{\mathbf{u}}^{a}_{i} := \mathbf{u}^{a}_{i}/u^{a}_{i}. $$
(6.16)
Thus
\(\hat{\mathbf{u}}^{a}_{i} (\mathbf{x}, t)\) denotes a unit vector in the direction of
\(P_{i}\) from
x at the latest time
\(\tau_{i} (\mathbf{x}, t)\) that information about
\(P_{i}\) reaches
\(\mathbf{x}_{i}\) at time
\(t\). From (
6.12) and (
6.15),
$$ \mathbf{v}_{i} \bigl(\tau_{i} (\mathbf{x}, t) \bigr) = \bigl(\beta_{i} (\mathbf{x}, t) \bigr)^{-1} \mathbf{v}^{a}_{i} (\mathbf{x}, t). $$
(6.17)
Equivalently,
$$ \mathbf{v}_{i} \circ\tau_{i} = \beta_{i}^{-1} \mathbf{v}^{a}_{i}. $$
(6.18)
This relation delivers the actual velocity at time
\(\tau_{i} ( \mathbf{x}, t)\) in terms of information available at
x and time
\(t\).
The spatial derivative (or ‘gradient’) of (
6.13) yields, with (
6.8)
$$ 2 \bigl(\nabla\mathbf{u}^{a}_{i} \bigr)^{T} \mathbf{u}^{a}_{i} = 2c^{2} (t-\tau _{i}) (-\nabla\tau_{i}) = -2cu^{a}_{i} \nabla\tau_{i}. $$
(6.19)
From (
6.6), the chain rule and (
6.9),
$$ \nabla\mathbf{u}^{a}_{i} = \partial\mathbf{x}_{i}/ \partial\tau \otimes\nabla\tau_{i} - \mathbf{1} = \dot{ \mathbf{x}}_{i} \otimes \nabla\tau_{i} - \mathbf{1} = \mathbf{v}_{i} \otimes\nabla\tau_{i} - \mathbf{1}, $$
(6.20)
whence
$$ \bigl(\nabla\mathbf{u}^{a}_{i} \bigr)^{T} = \nabla\tau_{i} \otimes\mathbf{v} _{i} - \mathbf{1}. $$
(6.21)
Hence (
6.19) may be written, noting (
6.16), as
$$ (\nabla\tau_{i} \otimes\mathbf{v}_{i} - \mathbf{1})\hat{ \mathbf{u}} ^{a}_{i} = -c\nabla\tau_{i}, $$
(6.22)
and so
$$ \bigl(\mathbf{v}_{i}\,.\, \hat{\mathbf{u}}^{a}_{i} + c \bigr) \nabla\tau_{i} = \hat{\mathbf{u}}^{a}_{i}. $$
(6.23)
Since, from (
6.17) and (
6.15)
2,
$$ \mathbf{v}_{i}\,.\, \hat{\mathbf{u}}^{a}_{i} + c = \beta^{-1}_{i} \mathbf{v}^{a}_{i} \,.\, \hat{\mathbf{u}}^{a}_{i} + c = \beta^{-1}_{i} \bigl( \mathbf{v}^{a}_{i}\,.\, \hat{\mathbf{u}}^{a}_{i} + \beta_{i} c \bigr) = \beta^{-1}_{i} c, $$
(6.24)
it follows from (
6.23) that
$$ \nabla\tau_{i} = \beta_{i} \hat{\mathbf{u}}^{a}_{i}/c. $$
(6.25)
From (
6.20)
3, (
6.25) and (
6.18),
$$ \nabla\mathbf{u}^{a}_{i} = \mathbf{v}_{i} \otimes\beta_{i}\, \hat{\mathbf{u}}^{a}_{i}/c - \mathbf{1} = \mathbf{v}^{a}_{i} \otimes \hat{ \mathbf{u}}^{a}_{i}/c - \mathbf{1}, $$
(6.26)
and, from (
6.8) and (
6.25),
$$ \nabla u^{a}_{i} = -c\nabla\tau_{i} = - \beta_{i} \hat{\mathbf{u}} ^{a}_{i}. $$
(6.27)
Further, from (
6.26) and (
6.15)
2,
$$ \mathrm{div}\, \mathbf{u}^{a}_{i} := \mathrm{tr} \bigl\{ \nabla\mathbf{u} ^{a}_{i} \bigr\} = \mathbf{v}^{a}_{i} \,.\, \hat{\mathbf{u}}^{a}_{i}/c - 3 = - \beta_{i} - 2. $$
(6.28)
In order to generalise the discussion of Sect.
5 to dynamic situations, taking account of signal transmission delay, it suffices to consider the potential functions: cf. Remark
5.6. A natural choice
\(\mathbf{A}^{d}_{w}\) for the
dynamic counterpart of
\(\mathbf{A}^{s}_{w}\) (cf. (
5.13)
2) would seem to be
$$ \mathbf{A}^{d}_{w} := \mu_{0}\, \sum _{i} q_{i} \, f \bigl(u^{a}_{i} \bigr) \mathbf{v}^{a}_{i} = \mu_{0} \sum _{i} q_{i}\, f \bigl(u^{a}_{i} \bigr) \beta_{i} \, \mathbf{v}_{i}. $$
(6.29)
Choice
\(\mathbf{A}^{d}_{w} (\mathbf{x}, t)\) takes account of the latest information concerning charge displacements and velocities that is available at
x and time
\(t\), has the form of (
5.13)
2, and would coincide with
\(\mathbf{A}^{s}_{w} (\mathbf{x}, t)\) were transmission times to be negligible. A choice of the corresponding dynamic electric potential might be
$$ \tilde{\psi}^{d}_{w} := \epsilon^{-1}_{0} \sum_{i}\, q_{i}\, f \bigl(u^{a} _{i} \bigr). $$
(6.30)
However, this field does not satisfy the counterpart of the Lorenz gauge relation (cf., e.g., Jackson [
13], p. 240), namely
$$ \partial\psi^{d}_{w}/\partial t + (1/\epsilon_{0} \mu_{0}) \mathrm{div}\, \mathbf{A}^{d}_{w} = 0. $$
(6.31)
To obtain an appropriate candidate
\(\psi^{d}_{w}\), consider (cf. (
6.29))
$$ \mathrm{div} \bigl\{ f \bigl(u^{a}_{i} \bigr) \mathbf{v}^{a}_{i} \bigr\} = \nabla f \bigl(u^{a}_{i} \bigr) \,.\, \mathbf{v}^{a}_{i} + f \bigl(u^{a}_{i} \bigr) \mathrm{div}\, \mathbf{v}^{a} _{i}. $$
(6.32)
From (
6.27),
$$ \nabla f \bigl(u^{a}_{i} \bigr) = f' \bigl(u^{a}_{i} \bigr) \nabla u^{a}_{i} = -\beta_{i}\, f' \bigl(u ^{a}_{i} \bigr)\hat{\mathbf{u}}^{a}_{i}, $$
(6.33)
while, from (
6.28),
$$ \mathrm{div}\, \mathbf{v}^{a}_{i} = \mathrm{div} \bigl\{ \partial \mathbf{u}^{a}_{i}/\partial t \bigr\} = \partial/ \partial t \bigl\{ \mathrm{div} \, \mathbf{u}^{a}_{i} \bigr\} = -\partial\beta_{i}/\partial t. $$
(6.34)
Accordingly, (
6.32), (
6.33) and (
6.34) imply
$$ \mathrm{div} \bigl\{ f \bigl(u^{a}_{i} \bigr) \mathbf{v}^{a}_{i} \bigr\} = -\beta_{i}\, f' \bigl(u ^{a}_{i} \bigr) \hat{ \mathbf{u}}^{a}_{i}\,.\,\mathbf{v}^{a}_{i} - \partial \beta_{i}/\partial t\, f \bigl(u^{a}_{i} \bigr). $$
(6.35)
From (
6.8) and (
6.15),
$$\begin{aligned} \partial/\partial t \bigl\{ f \bigl(u^{a}_{i} \bigr) \bigr\} =& f' \bigl(u^{a}_{i} \bigr) \partial u ^{a}_{i}/\partial t = f' \bigl(u^{a}_{i} \bigr) c(1-\partial\tau_{i}/\partial t) \\ =& f' \bigl(u^{a}_{i} \bigr) cc^{-1} \hat{\mathbf{u}}^{a}_{i}\,.\, \mathbf{v} ^{a}_{i} = f' \bigl(u^{a}_{i} \bigr) \hat{\mathbf{u}}^{a}_{i}\,.\, \mathbf{v}^{a}_{i}. \end{aligned}$$
(6.36)
Hence, from (
6.35) and (
6.36),
$$ \mathrm{div} \bigl\{ f \bigl(u^{a}_{i} \bigr) \mathbf{v}^{a}_{i} \bigr\} = -\partial/\partial t \bigl\{ \beta_{i}\, f \bigl(u^{a}_{i} \bigr) \bigr\} , $$
(6.37)
and thus
$$ \psi^{d}_{w} := \epsilon^{-1}_{0} \sum _{i}\, q_{i} \beta_{i}\, f \bigl(u ^{a}_{i} \bigr) $$
(6.38)
satisfies (
6.31).
The results of Sect.
5 can now be generalised by essentially replacing superscript ‘
\(s\)’ by ‘
\(d\)’. Specifically, function
\(f\) is known once weighting function
\(w\) is chosen: this follows from Theorem
3.1, consequent upon definition (
3.5), (
3.31)
2 and (
3.51). In particular (cf. Remark
3.6),
\(f(u) = 0\) if
\(u < \epsilon\) and
\(f(u) = 1/4\pi u\) if
\(u > \epsilon+ \delta\). Knowledge of
\(f\), and the latest information concerning displacements and velocities (namely
\(\mathbf{u}^{a}_{i}\) and
\(\mathbf{v}^{a}_{i}\)), yield fields
\(\mathbf{A}^{d}_{w}\) and
\(\psi^{d}_{w}\) via (
6.29) and (
6.38). The corresponding dynamic electric field
\(\mathbf{E}^{d}_{w}\) and magnetic field
\(\mathbf{B}^{d}_{w}\) are
$$ \mathbf{E}^{d}_{w} := -\nabla\psi^{d}_{w} - \partial\mathbf{A}^{d} _{w}/\partial t, \qquad \mathbf{B}^{d}_{w} := \mathrm{curl}\, \mathbf{A}^{d}_{w}. $$
(6.39)
8 Summary and Concluding Remarks
The microscopic basis of classical macroscopic electromagnetic relations has been investigated via recognition of the rôle played by spatial and temporal averaging, here implemented in terms of weighting functions. Modelling electrons and nuclei as point charges, any choice
\(w\) of weighting function yielded definitions
\(\rho_{w}\) and
\(\mathbf{j}_{w}\) of charge and current densities which satisfy (
1.6). Any solution
a to
\(\mathrm{div}\, \mathbf{a} = w\) gave rise to purely electrokinetic fields
\(\mathbf{D}_{w}\) and
\(\mathbf{H}_{w}\) which satisfy (
1.1) and (
1.2). At any prescribed scale
\(\epsilon\), choice of a specific and natural weighting function resulted in binary decompositions (
3.35) of
\(\mathbf{D}_{w}\) and
\(\mathbf{H}_{w}\). These decompositions introduced electric and magnetic polarisation densities together with electrokinetic fields
\(\boldsymbol {\mathcal {E}}_{w}\) and
\(\boldsymbol {\mathcal {B}} _{w}\) that were expressible in terms of scalar and vector potentials
\(\psi_{w}\) and
\(\boldsymbol {\mathcal {A}}_{w}\), respectively. The experimental laws of Coulomb and Biot-Savart enabled the macroscopic time-independent electric and magnetic fields
\(\mathbf{E}^{s}_{w}\) and
\(\mathbf{B}^{s} _{w}\) in free space to be expressed as scalar multiples of
\(\boldsymbol {\mathcal {E}} _{w}\) and
\(\boldsymbol {\mathcal {B}}^{s}_{w}\) and hence in terms of potential functions
\(\psi^{s}_{w}\) and
\(\mathbf{A}^{s}_{w}\). Fields
\(\mathbf{B} ^{s}_{w}\) and
\(\mathbf{A}^{s}_{w}\) satisfy (
1.3) and (
1.5), and
\(\mathbf{E}^{s}_{w}\) satisfies (
1.7) and (
1.8) in this macroscopically stationary context. Generalisation to macroscopically dynamic situations was effected in two stages. In the first stage the dynamic electric field was formally defined via (
1.7), with
\(\psi\) and
A identified with
\(\psi^{s}_{w}\) and
\(\mathbf{A}^{s}_{w}\), and the consequences exhibited in Theorem
5.1. The second stage addressed the consequences of non-instantaneous transmission of information: this involved selection of appropriate modified versions
\(\psi^{d}_{w}\) and
\(\mathbf{A}^{d}_{w}\) of
\(\psi^{s}_{w}\) and
\(\mathbf{A}^{s}_{w}\), and resulted in the final form of the relations under investigation, displayed in Theorem
6.1. The individual contribution of any given charge to the force on another moving charge, as predicted by the Lorentz relation (
1.9), was computed as an exercise and for completeness.